Large deviations for distributions of sums of random variables: Markov chain method

  • Authors:

  • V. R. Fatalov

  • Affiliations:

  • Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

  • Venue:
  • Problems of Information Transmission
  • Year:
  • 2010

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Abstract

Let {驴 k } k=0 驴 be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $$ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) , n 驴 驴, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| p , p 0, and exponential random variables with g(x) = x for x 驴 0.